I don't know how much it would effect the outcome but M class stars are entirely convective so they will eventually fuse practically all their Hydrogen and continue as a Helium burning convective star. But one thing is the rate they fuse at is so much slower than the core fusion stars, in order of 100s of billions of years, perhaps into the trillions. They also happen to be quite common.

I calculated that I've spent just over a week airborne over the past 12 months covering a distance of about half a light second.  Does this also mean that I've travelled half a second further into the future than if I'd remained at home, or is the maths more complicated?

There's another thing to consider: latitude.  It's perhaps the most important variable?

I. SETTING THE SCENE
Sam is in the office and has just finished reading Plato’s Meno1 in which Socrates uses a self-discovery technique to teach a boy Pythagoras’ theorem. Sam is inspired by this dialog and is pondering its applicability to lecturing undergraduate physics when a tap on the door breaks that chain of thought. Kim enters the room looking bleary eyed and pale. “Been out celebrating the last lecture of the year” Sam surmises, little knowing that other things have kept Kim awake.

II. THE DIALOG
Kim:
Your lectures on special relativity fascinated me, and when I got home I wondered if I could construct a simple experiment to prove or disprove time dilation, the aspect of special relativity that interests me the most. While lying in bed before dozing off I realised that a clock placed at the equator should run slower than a clock placed at the pole. So I did a little calculation and found that special relativity predicts that a clock on the equator runs slower by about 100 nanoseconds per day with respect to a clock at the pole. While this effect is not large it is certainly measurable with modern atomic clocks. So I went onto the internet to see if I could find any reference to such an experiment and to my surprise I couldn’t.

I was starting to get so frustrated that I couldn’t sleep. I glanced at the clock (3am). I thought to myself “How accurate is my clock? I should check it against internet time.” Then it occurred to me that the world timing standard organisations must mention a latitude effect on local clock accuracies. So I got onto the internet again and checked The Bureau International des Poids et Mesures (BIPM)2 as they calculate the international atomic time (TAI). BIPM calculate TAI from atomic clocks located in more than 30 countries around the world. I was sure that I must find something about the latitude effect on their web site. After spending hours trawling through the site and then other sites on the web, I came up with nothing. There was a discussion of the relativistic effect of placing clocks at high altitudes, but nothing about latitude. In my despair I gave up and collapsed into a fitful sleep. I came to see you today in the hope that you could cure my insomnia.

If a ship is in a circular orbit at exactly 1.5 times the planet's radius (this is 3185km altitude for Earth), then its clocks will tick at exactly the same rate as clocks on the planet surface.

I suspected it was more complicated, but not that the altitude would more than cancel it, since 10km is not much at the scale of the earth.

However, it is more complicated than that.  The mass (or density) of Earth also matters.  A less dense Earth would make the 10km gain in altitude less significant.  If Earth was the same size but 1/3 of the mass, then the special and general relativistic effects would have more neatly cancelled on your air travel.  (Of course a 1/3 as massive Earth might change how air travel works, among other problems).

For a planet and moon system with equal density, isn't that coincidentally also the Roche limit?

I came across an interesting video. I am wondering how could I calculate the orbital period of the configuration at 3:32. I know all of these are three body orbits which are unstable, but the one I am interested in appears to be doable in Space Engine.
[youtube]8_RRZcqBEAc[/youtube]

Edit:

► Show Spoiler

Sitnikov problem. What would be the period of m3, assuming it is of planetary mass and m1 and m2 are stars? (It looks like m1=m2, at least according to the diagram.

...star formation would cease even if astronomical amounts of Hydrogen would still be present in the darkness of the Universe.

While playing around with time dilation formulas, I just came across something interesting.

If a ship is in a circular orbit at exactly 1.5 times the planet's radius (this is 3185km altitude for Earth), then its clocks will tick at exactly the same rate as clocks on the planet surface.  For a closer orbit, special relativity dominates and the orbiting clock will tick slower due to its speed, and for higher orbits general relativity dominates and it ticks faster due to the weaker gravity.

So for astronauts on the ISS (about 400km altitude), time is slower for them (by about 25 microseconds per day!) while for orbiting GPS satellites (roughly circular orbits at about 20,000km altitude), time is faster than on Earth (by about 38 microseconds per day).  This actually must be accounted for in order for GPS to work properly, since the system requires extremely precise time measurements to pinpoint your location to within a few meters.  Light travels about 0.3 meters per nanosecond, so if GPS did not account for time dilation then it would accumulate an error of 11 kilometers per day!

These corrections were actually built into the global positioning system from the outset, so the fact that GPS did not fail immediately is a nice proof that time dilation is real.